Theorem redcwlpo 17239

Description: Decidability of real number equality implies the Weak Limited Principle of Omniscience (WLPO). We expect that we'd need some form of countable choice to prove the converse.

Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 17238). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones.

Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO".

WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10690 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)

Assertion

Ref	Expression
redcwlpo	|- ( A. x e. RR A. y e. RR DECID x = y -> om e. WOmni )

Distinct variable group:   x, y

Proof of Theorem redcwlpo

Dummy variables f i j z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . 6 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> A. x e. RR A. y e. RR DECID x = y )
2		elmapi 6944	. . . . . . . . 9 |- ( f e. ( { 0 , 1 } ^m NN ) -> f : NN --> { 0 , 1 } )
3	2	adantl 277	. . . . . . . 8 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> f : NN --> { 0 , 1 } )
4		oveq2 6093	. . . . . . . . . . 11 |- ( i = j -> ( 2 ^ i ) = ( 2 ^ j ) )
5	4	oveq2d 6101	. . . . . . . . . 10 |- ( i = j -> ( 1 / ( 2 ^ i ) ) = ( 1 / ( 2 ^ j ) ) )
6		fveq2 5695	. . . . . . . . . 10 |- ( i = j -> ( f ` i ) = ( f ` j ) )
7	5, 6	oveq12d 6103	. . . . . . . . 9 |- ( i = j -> ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = ( ( 1 / ( 2 ^ j ) ) x. ( f ` j ) ) )
8	7	cbvsumv 12146	. . . . . . . 8 |- sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = sum_ j e. NN ( ( 1 / ( 2 ^ j ) ) x. ( f ` j ) )
9	3, 8	trilpolemcl 17220	. . . . . . 7 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) e. RR )
10		1red 8342	. . . . . . 7 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> 1 e. RR )
11		eqeq1 2245	. . . . . . . . 9 |- ( x = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) -> ( x = y <-> sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y ) )
12	11	dcbid 850	. . . . . . . 8 |- ( x = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) -> (DECID x = y <-> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y ) )
13		eqeq2 2248	. . . . . . . . 9 |- ( y = 1 -> ( sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y <-> sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
14	13	dcbid 850	. . . . . . . 8 |- ( y = 1 -> (DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y <-> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
15	12, 14	rspc2v 2943	. . . . . . 7 |- ( ( sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) e. RR /\ 1 e. RR ) -> ( A. x e. RR A. y e. RR DECID x = y -> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
16	9, 10, 15	syl2anc 415	. . . . . 6 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> ( A. x e. RR A. y e. RR DECID x = y -> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
17	1, 16	mpd 13	. . . . 5 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 )
18	3, 8	redcwlpolemeq1 17238	. . . . . 6 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> ( sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 <-> A. z e. NN ( f ` z ) = 1 ) )
19	18	dcbid 850	. . . . 5 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> (DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 <-> DECID A. z e. NN ( f ` z ) = 1 ) )
20	17, 19	mpbid 147	. . . 4 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> DECID A. z e. NN ( f ` z ) = 1 )
21	20	ralrimiva 2623	. . 3 |- ( A. x e. RR A. y e. RR DECID x = y -> A. f e. ( { 0 , 1 } ^m NN )DECID A. z e. NN ( f ` z ) = 1 )
22		nnex 9313	. . . 4 |- NN e. _V
23		iswomninn 17234	. . . 4 |- ( NN e. _V -> ( NN e. WOmni <-> A. f e. ( { 0 , 1 } ^m NN )DECID A. z e. NN ( f ` z ) = 1 ) )
24	22, 23	ax-mp 5	. . 3 |- ( NN e. WOmni <-> A. f e. ( { 0 , 1 } ^m NN )DECID A. z e. NN ( f ` z ) = 1 )
25	21, 24	sylibr 134	. 2 |- ( A. x e. RR A. y e. RR DECID x = y -> NN e. WOmni )
26		nnenom 10886	. . 3 |- NN ~~ om
27		enwomni 7511	. . 3 |- ( NN ~~ om -> ( NN e. WOmni <-> om e. WOmni ) )
28	26, 27	ax-mp 5	. 2 |- ( NN e. WOmni <-> om e. WOmni )
29	25, 28	sylib 122	1 |- ( A. x e. RR A. y e. RR DECID x = y -> om e. WOmni )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   {cpr 3710   class class class wbr 4130   omcom 4737   -->wf 5373   ` cfv 5377  (class class class)co 6085   ^m cmap 6922   ~~ cen 7020  WOmnicwomni 7504   RRcr 8179   0cc0 8180   1c1 8181   x. cmul 8185   / cdiv 9005   NNcn 9307   2c2 9358   ^cexp 10990   sum_csu 12138

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299  ax-caucvg 8300

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-womni 7505  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-n0 9569  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-ico 10307  df-fz 10423  df-fzo 10561  df-seqfrec 10900  df-exp 10991  df-ihash 11231  df-cj 11623  df-re 11624  df-im 11625  df-rsqrt 11780  df-abs 11781  df-clim 12064  df-sumdc 12139

This theorem is used by: (None)